Transition in plane Poiseuille flow with a streamwise rotation

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Transition in plane Poiseuille flow with a
stream-wise rotation
11th EUROMECH European Turbulence
Conference Porto, Portugal 25 28 June 2007

• M. Nagata and S. Masuda
• Department of Aeronautics and Astronautics,
  Graduate School of Engineering
• and
• Advanced Research Institute of Fluid Science and
  Engineering
• Kyoto University, JAPAN

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Background
The basic flow , which is the exact solution of
the Navier-Stokes equation, has no
span-wise component. The Coriolis force does not
act in the span-wise direction.
Recktenwald , Ch. Brucker , W. Schröder (2004)
Experiment Oberlack(2006) DNS
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Application
Instability of Meridian flow across the equator
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Organisation

• 1. Mathematical formulation
• 2. Linear stability analysis
• 3. Nonlinear analysis
• 4. Summary

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Non-dimensional equations
O rotation number,
Reynolds number,
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Non-linear perturbation equation
perturbation to the basic flow
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Non-linear perturbation equation
Operating
on momentum eq.
where
Averaging the x- and y-compoments of
Navier-Stokes equations
.
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Non-linear perturbation equation
Non-slip boundary condition
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Organisation

• 1. Mathematical formulation
• 2. Linear stability analysis
• 3. Nonlinear analysis
• 4. Summary

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Linear stability analysis
Neglect the non-linear terms
Non-slip boundary condition
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Linear analysis
Normal mode
collocation points
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Result Linear Stability Analysis
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Linear analysis
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Linear analysis
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Neutral curves

2D instablities set in first Stabilising effect
of rotation
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Neutral curves
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Organisation

• 1. Mathematical formulation
• 2. Linear stability analysis
• 3. Nonlinear analysis
• 4. Summary

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Nonlinear perturbation equations
where
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The non-linear algebraic equation

• collocation points

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Result
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Momentum transport
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Phase velocity
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Mean Flow
z
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Flow field in the cross section
R90
Snap shots
colour code streamwise velocity
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Flow field in the cross section R70
Colour code stream-wise vorticity
3D structure of the stream-wise vorticity
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Streamwise vorticity
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Organisation

• 1. Mathematical formulation
• 2. Linear stability analysis
• 3. Nonlinear analysis
• 4. Summary

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Conclusion and Future work

• It is found that the basic flow loses its
  stability at a low Reynolds number, far below the
  critical Reynolds number for the case without
  rotation, and a flow with a three-dimensional
  structure bifurcates.
• Nonlinear analysis exhibits, in particular, that
  the bifurcating three-dimensional solution
  generates a momentum transport in the spanwise
  direction.
• Stability of the nonlinear states will be
  analysed in future. (Preliminary results in
  Appendix)

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Appendix Stability of nonlinear solutions